DLMF:15.8.E22 (Q5079)

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DLMF:15.8.E22
No description defined

    Statements

    Defining formula
    F ⁡ ( a , b 1 2 ⁢ ( a + b + 1 ) ; z ) = ( 1 - z - 1 - 1 1 - z - 1 + 1 ) a ⁢ F ⁡ ( a , 1 2 ⁢ ( a + b ) a + b ; 4 ⁢ 1 - z - 1 ( 1 - z - 1 + 1 ) 2 ) , Gauss-hypergeometric-F 𝑎 𝑏 1 2 𝑎 𝑏 1 𝑧 superscript 1 superscript 𝑧 1 1 1 superscript 𝑧 1 1 𝑎 Gauss-hypergeometric-F 𝑎 1 2 𝑎 𝑏 𝑎 𝑏 4 1 superscript 𝑧 1 superscript 1 superscript 𝑧 1 1 2 {\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=% \left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F\left({a,\tfrac{1% }{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}% }\right),}}
    0 references
    DLMF id
    DLMF:15.8.E22
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    constraint
    | ph ⁡ ( - z ) | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi}}
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    ℜ ⁡ z < 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z<\frac{1}{2}}}
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    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2akdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aldec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1udec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1tdec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1rdec
    0 references
     
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